Subgroups — #25101546

Order 11008.

714 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19143231	C₂	1	1	yes	maximal, characteristic
2	C₂×#14582961	C₂	1	1	yes	maximal, characteristic
3	#19143012	C₂	1	1	yes	maximal, characteristic
4	#19135369	C₂	1	2	yes	maximal, complemented
5	#19135367	C₂	1	2	yes	maximal, complemented
6	#19134801	C₂	1	2	yes	maximal
7	#19134798	C₂	1	2	yes	maximal
8	#19134501	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#13063260	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×#13063260	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₃×#1778	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×M₁₆×D₄₃	C₂²	1	2	yes	—
13	C₂×#13063282	C₂²	1	1	yes	characteristic
14	#14582961	—	2	2	no	—
15	#14582961	C₄	1	8	yes	—
16	#14582958	—	2	2	no	—
17	C₂×#13063261	C₂²	1	2	yes	—
18	C₂×#13063260	C₂²	1	2	yes	—
19	C₂×#13063260	C₂²	1	2	yes	—
20	#14582922	C₂²	1	2	yes	—
21	C₂×Dic₄₃⋊C₈	C₂²	1	1	yes	characteristic
22	C₂×Dic₄₃⋊C₈	C₂²	1	1	yes	characteristic
23	#14582862	C₂²	1	2	yes	—
24	C₂×(C₂×D₄₃)⋊C₈	C₂²	1	1	yes	characteristic
25	C₂×(C₂×D₄₃)⋊C₈	C₂²	1	1	yes	characteristic
26	#14582856	—	2	2	no	—
27	#14582853	—	2	2	no	—
28	C₄×#10621877	C₂²	1	1	yes	characteristic
29	C₄×#10621877	C₂²	1	1	yes	characteristic
30	M₁₆×Dic₄₃	C₂²	1	2	yes	—
31	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
32	#14582664	C₂²	1	2	yes	—
33	#14582663	C₂²	1	2	yes	—
34	#14582663	C₂²	1	2	yes	—
35	C₂×#108×C₄₃	C₂²	1	2	yes	—
36	C₄×M₁₆×C₄₃	C₂²	1	1	yes	characteristic
37	C₄×M₁₆×C₄₃	C₂²	1	1	yes	characteristic
38	C₂×#97×C₄₃	C₂²	1	1	yes	characteristic
39	C₄₃×#341	C₂²	1	2	yes	—
40	C₂²×C₄×D₄₃	C₂³	1	1	yes	characteristic
41	C₂×#10621877	C₂³	1	2	yes	—
42	C₂×#10621877	C₂³	1	2	yes	—
43	C₂×#10621872	—	2	2	no	—
44	M₁₆×D₄₃	—	2	4	no	—
45	M₁₆×D₄₃	—	2	4	no	—
46	C₂²×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
47	C₂²×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
48	C₂×C₈×D₄₃	—	2	2	no	—
49	#13063282	C₂×C₄	1	4	yes	—
50	#13063282	D₄	1	2	yes	—
51	#13063282	D₄	1	2	yes	—
52	C₂×Dic₄₃⋊C₄	Q₈	1	2	yes	—
53	C₂×(C₂×D₄₃)⋊C₄	Q₈	1	2	yes	—
54	C₂×C₄×Dic₄₃	C₂³	1	1	yes	characteristic
55	#13063261	—	2	4	no	—
56	#13063260	—	2	4	no	—
57	#13063260	—	2	4	no	—
58	Dic₄₃⋊C₈	—	2	4	no	—
59	Dic₄₃⋊C₈	C₂×C₄	1	4	yes	—
60	Dic₄₃⋊C₈	C₂×C₄	1	4	yes	—
61	(C₂×D₄₃)⋊C₈	—	2	4	no	—
62	(C₂×D₄₃)⋊C₈	C₂×C₄	1	4	yes	—
63	(C₂×D₄₃)⋊C₈	C₂×C₄	1	4	yes	—
64	#13063236	—	2	1	no	—
65	#13063236	—	2	1	no	—
66	#13063235	—	2	2	no	—
67	C₄×C₄₃⋊C₈	—	2	1	no	—
68	C₄×C₄₃⋊C₈	—	2	1	no	—
69	C₄×C₄₃⋊C₈	C₂×C₄	1	4	yes	—
70	C₈×Dic₄₃	—	2	2	no	—
71	C₂×M₁₆×C₄₃	C₂³	1	2	yes	—
72	C₂×M₁₆×C₄₃	C₂³	1	2	yes	—
73	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
74	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
75	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
76	#108×C₄₃	—	2	4	no	—
77	#97×C₄₃	—	2	1	no	—
78	#97×C₄₃	—	2	1	no	—
79	#97×C₄₃	C₂×C₄	1	4	yes	—
80	C₄×C₈×C₄₃	—	2	1	no	—
81	C₄×C₈×C₄₃	—	2	1	no	—
82	C₂³×D₄₃	C₂×Q₈	1	1	yes	characteristic
83	C₂²×C₄×C₄₃	C₂×Q₈	1	2	yes	—
84	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
85	C₂²×Dic₄₃	C₂×Q₈	1	1	yes	characteristic
86	C₂²×Dic₄₃	C₂×Q₈	1	2	yes	—
87	C₂×C₄×D₄₃	—	2	2	no	—
88	C₂×C₄×D₄₃	—	2	2	no	—
89	C₂×C₄×D₄₃	C₂²×C₄	1	2	yes	—
90	C₂×C₄×D₄₃	C₂×D₄	1	2	yes	—
91	M₁₆×C₄₃	—	2	2	no	—
92	M₁₆×C₄₃	—	2	2	no	—
93	M₁₆×C₄₃	—	2	2	no	—
94	M₁₆×C₄₃	#41	1	4	yes	—
95	C₂×C₈×C₄₃	—	2	2	no	—
96	C₂×C₈×C₄₃	—	2	2	no	—
97	C₂×C₈×C₄₃	—	2	2	no	—
98	C₂×C₈×C₄₃	—	2	2	no	—
99	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
100	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
101	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
102	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
103	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
104	(C₂×D₄₃)⋊C₄	C₄⋊C₄	1	4	yes	—
105	(C₂×D₄₃)⋊C₄	C₄⋊C₄	1	4	yes	—
106	Dic₄₃⋊C₄	C₄⋊C₄	1	4	yes	—
107	Dic₄₃⋊C₄	C₄⋊C₄	1	4	yes	—
108	C₄×Dic₄₃	C₂²×C₄	1	2	yes	—
109	C₄×Dic₄₃	C₂×D₄	1	2	yes	—
110	#10621877	—	2	2	no	—
111	#10621877	—	2	2	no	—
112	#10621877	—	2	2	no	—
113	#10621877	#41	1	4	yes	—
114	C₂×C₄₃⋊C₈	—	2	2	no	—
115	C₂×C₄₃⋊C₈	—	2	2	no	—
116	C₂×C₄₃⋊C₈	—	2	2	no	—
117	C₂×C₄₃⋊C₈	—	2	2	no	—
118	C₂×C₄₃⋊C₈	C₂²×C₄	1	2	yes	—
119	C₂×C₄₃⋊C₈	C₂²×C₄	1	2	yes	—
120	#10621872	—	2	8	no	—
121	C₈×D₄₃	—	2	8	no	—
122	C₂³×C₄₃	C₂²×Q₈	1	1	yes	characteristic
123	C₂²×D₄₃	—	2	2	no	—
124	C₂²×D₄₃	—	4	2	no	—
125	C₂²×D₄₃	C₂×C₄⋊C₄	1	2	yes	—
126	C₂×C₄×C₄₃	—	4	2	no	—
127	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
128	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
129	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
130	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
131	C₂×C₄×C₄₃	C₂×#41	1	2	yes	—
132	C₂×C₄×C₄₃	C₂×#41	1	2	yes	—
133	C₂×Dic₄₃	—	2	2	no	—
134	C₂×Dic₄₃	—	4	2	no	—
135	C₂×Dic₄₃	C₂×C₄⋊C₄	1	4	yes	—
136	C₂×Dic₄₃	C₂×C₄⋊C₄	1	2	yes	—
137	C₄×D₄₃	—	2	2	no	—
138	C₄×D₄₃	—	2	2	no	—
139	C₄×D₄₃	—	2	2	no	—
140	C₄×D₄₃	—	2	2	no	—
141	C₈×C₄₃	—	2	2	no	—
142	C₈×C₄₃	—	2	2	no	—
143	C₈×C₄₃	—	2	4	no	—
144	C₈×C₄₃	—	4	2	no	—
145	C₄₃⋊C₈	—	2	2	no	—
146	C₄₃⋊C₈	—	2	2	no	—
147	C₄₃⋊C₈	—	2	4	no	—
148	C₄₃⋊C₈	—	4	2	no	—
149	#21372	—	43	1	no	maximal, Sylow
150	C₂²×C₄₃	—	2	2	no	—
151	C₂²×C₄₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
152	C₂²×C₄₃	#533	1	2	yes	—
153	C₂×D₄₃	—	2	4	no	—
154	C₂×D₄₃	—	2	2	no	—
155	C₂×D₄₃	—	8	2	no	—
156	C₄×C₄₃	—	2	2	no	—
157	C₄×C₄₃	—	2	2	no	—
158	C₄×C₄₃	C₄×#41	1	1	yes	characteristic
159	C₄×C₄₃	C₄×#41	1	1	yes	characteristic
160	C₄×C₄₃	#525	1	2	yes	—
161	Dic₄₃	—	2	4	no	—
162	Dic₄₃	—	2	2	no	—
163	#2921	—	43	1	no	—
164	C₂×#433	—	43	1	no	—
165	#2873	—	43	1	no	—
166	#1950	—	43	2	no	—
167	#1950	—	43	2	no	—
168	#1855	—	43	2	no	—
169	#1855	—	43	2	no	—
170	#1778	—	43	1	no	—
171	#1778	D₄₃	1	1	yes	characteristic, complemented
172	C₄×#108	—	43	1	no	—
173	C₄×#108	—	43	1	no	—
174	C₂×C₄₃	—	2	2	no	—
175	C₂×C₄₃	#2296	1	1	yes	characteristic
176	C₂×C₄₃	#2916	1	2	yes	—
177	D₄₃	—	4	2	no	—
178	C₂²×M₁₆	—	43	2	no	—
179	C₂×#117	—	43	1	no	—
180	#433	—	43	8	no	—
181	#433	—	86	2	no	—
182	#433	—	86	2	no	—
183	C₂×#108	—	43	2	no	—
184	C₂×#108	—	43	2	no	—
185	C₂×#108	—	43	2	no	—
186	C₂×#108	C₂×D₄₃	1	2	yes	—
187	#423	—	43	2	no	—
188	C₂×C₄⋊C₈	—	43	1	no	—
189	C₂×C₄⋊C₈	—	43	1	no	—
190	#407	—	43	2	no	—
191	C₂×C₂²⋊C₈	—	43	1	no	—
192	C₂×C₂²⋊C₈	—	43	1	no	—
193	#405	—	86	2	no	—
194	#405	—	86	2	no	—
195	C₄×M₁₆	—	43	2	no	—
196	C₄×M₁₆	—	43	1	no	—
197	C₄×M₁₆	—	43	1	no	—
198	C₄×M₁₆	C₂×D₄₃	1	1	yes	characteristic
199	C₄×M₁₆	C₂×D₄₃	1	1	yes	characteristic
200	C₂×#97	C₂×D₄₃	1	1	yes	characteristic
201	C₂×C₄×C₈	—	43	1	no	—
202	#341	—	43	2	no	—
203	#341	—	43	2	no	—
204	#341	—	43	2	no	—
205	#341	C₂×D₄₃	1	2	yes	—
206	C₄₃	#21372	1	1	yes	characteristic, complemented, Sylow
207	C₂³×C₄	—	43	1	no	—
208	C₂×M₁₆	—	43	2	no	—
209	C₂×M₁₆	—	43	2	no	—
210	C₂×M₁₆	—	86	2	no	—
211	C₂×M₁₆	—	86	4	no	—
212	C₂×M₁₆	—	86	4	no	—
213	C₂×M₁₆	C₂²×D₄₃	1	2	yes	—
214	C₂×M₁₆	C₂²×D₄₃	1	2	yes	—
215	C₂²×C₈	—	43	1	no	—
216	C₂²×C₈	—	43	1	no	—
217	C₂²×C₈	—	86	2	no	—
218	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
219	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
220	#117	—	43	4	no	—
221	#117	—	43	2	no	—
222	#117	—	43	2	no	—
223	C₂×C₄⋊C₄	—	43	2	no	—
224	C₂×C₂²⋊C₄	—	43	2	no	—
225	C₂×C₄²	—	43	1	no	—
226	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
227	#108	—	2	4	no	—
228	#108	—	86	4	no	—
229	#108	—	86	4	no	—
230	#108	—	86	4	no	—
231	C₄⋊C₈	—	43	4	no	—
232	C₄⋊C₈	—	43	4	no	—
233	C₄⋊C₈	—	86	4	no	—
234	C₂²⋊C₈	—	43	4	no	—
235	C₂²⋊C₈	—	43	4	no	—
236	C₂²⋊C₈	—	86	4	no	—
237	#97	—	2	1	no	—
238	#97	—	2	1	no	—
239	#97	—	86	2	no	—
240	#97	—	86	1	no	—
241	#97	—	86	1	no	—
242	#97	C₄×D₄₃	1	4	yes	—
243	C₄×C₈	—	2	1	no	—
244	C₄×C₈	—	2	1	no	—
245	C₄×C₈	—	43	4	no	—
246	C₄×C₈	—	86	2	no	—
247	C₄×C₈	—	86	1	no	—
248	C₄×C₈	—	86	1	no	—
249	C₂⁴	—	43	1	no	—
250	C₂²×C₄	—	43	1	no	—
251	C₂²×C₄	—	43	2	no	—
252	C₂²×C₄	—	43	2	no	—
253	C₂²×C₄	—	43	2	no	—
254	C₂²×C₄	—	86	2	no	—
255	C₂²×C₄	—	86	2	no	—
256	C₂²×C₄	Q₈×D₄₃	1	2	yes	—
257	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
258	M₁₆	—	2	2	no	—
259	M₁₆	—	2	2	no	—
260	M₁₆	—	2	2	no	—
261	M₁₆	—	43	4	no	—
262	M₁₆	—	86	8	no	—
263	M₁₆	—	86	2	no	—
264	M₁₆	—	86	2	no	—
265	M₁₆	—	86	2	no	—
266	M₁₆	#10621898	1	4	yes	—
267	C₂×C₈	—	2	2	no	—
268	C₂×C₈	—	2	2	no	—
269	C₂×C₈	—	2	2	no	—
270	C₂×C₈	—	2	2	no	—
271	C₂×C₈	—	43	2	no	—
272	C₂×C₈	—	43	2	no	—
273	C₂×C₈	—	86	8	no	—
274	C₂×C₈	—	86	2	no	—
275	C₂×C₈	—	86	2	no	—
276	C₂×C₈	—	86	2	no	—
277	C₂×C₈	—	86	2	no	—
278	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
279	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
280	C₄⋊C₄	—	43	4	no	—
281	C₄⋊C₄	—	43	4	no	—
282	C₂²⋊C₄	—	43	4	no	—
283	C₂²⋊C₄	—	43	4	no	—
284	C₄²	—	43	2	no	—
285	C₄²	—	43	2	no	—
286	C₄²	C₂×C₄×D₄₃	1	2	yes	—
287	C₄²	D₄×D₄₃	1	1	yes	characteristic
288	C₄²	D₄×D₄₃	1	1	yes	characteristic
289	C₂³	—	43	2	no	—
290	C₂³	—	86	2	no	—
291	C₂³	—	172	2	no	—
292	C₂³	C₂×Q₈×D₄₃	1	1	yes	characteristic
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	43	4	no	—
295	C₂×C₄	—	43	2	no	—
296	C₂×C₄	—	86	2	no	—
297	C₂×C₄	—	86	2	no	—
298	C₂×C₄	—	86	2	no	—
299	C₂×C₄	—	86	2	no	—
300	C₂×C₄	—	86	2	no	—
301	C₂×C₄	—	172	2	no	—
302	C₂×C₄	C₄⋊C₄×D₄₃	1	2	yes	—
303	C₂×C₄	C₄⋊C₄×D₄₃	1	2	yes	—
304	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
305	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
306	C₂×C₄	#41×D₄₃	1	2	yes	—
307	C₂×C₄	C₂×#10621898	1	2	yes	—
308	C₈	—	2	2	no	—
309	C₈	—	2	2	no	—
310	C₈	—	2	4	no	—
311	C₈	—	4	2	no	—
312	C₈	—	86	2	no	—
313	C₈	—	86	2	no	—
314	C₈	—	86	4	no	—
315	C₈	—	172	2	no	—
316	C₂²	—	2	2	no	—
317	C₂²	—	86	4	no	—
318	C₂²	—	86	2	no	—
319	C₂²	—	344	2	no	—
320	C₂²	C₂×C₄⋊C₄×D₄₃	1	1	yes	characteristic
321	C₂²	#14583453	1	2	yes	—
322	C₄	—	2	2	no	—
323	C₄	—	2	2	no	—
324	C₄	—	86	4	no	—
325	C₄	—	86	2	no	—
326	C₄	#14583367	1	1	yes	characteristic
327	C₄	C₄×#10621898	1	1	yes	characteristic
328	C₄	#14583408	1	2	yes	—
329	C₂	—	2	2	no	—
330	C₂	—	172	2	no	—
331	C₂	#19137075	1	1	yes	characteristic
332	C₂	#19143205	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₄₃×#1778	C₂
socle	C₂²×C₄₃	C₂²×C₄⋊C₄
prime radical	C₂³×D₄₃	C₂×Q₈
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	#1778	D₄₃
hypocenter	C₄₃	#21372
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1778

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
4. #19135369	C₂	semidirect
5. #19135367	C₂	semidirect
8. #19134501	C₂	semidirect
9. C₄×#13063260	C₂	semidirect
10. C₄×#13063260	C₂	semidirect
11. C₄₃×#1778	C₂	semidirect
171. #1778	D₄₃	semidirect
206. C₄₃	#21372	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#19143231	2
C₂×#14582961	2
#19143012	2
#19135369	2
#19135367	2
#19134801	2
#19134798	2
#19134501	2
C₄×#13063260	2
C₄₃×#1778	2
#21372	43